
Obtain a balanced binary tree decomposition of the input
treewidth \(k\)  graph \(G\)  in Logspace via Proposition REF  [1]}.

Obtain the tree decomposition of \(G^{\prime }\)  (as required by Proposition REF 
and specified by Lemma REF ) from the tree decomposition of \(G\) .

Perform a chordal completion of \(G^{\prime }\)  by adding edges to every bag.

Obtain a PEO tree of \(G^{\prime }\)  of height \(O(\log {n})\) , where every vertex has at most \(k\) 
children via Lemma REF .

Construct a NLC width \((k+2)\)  expression for \(L(G^{\prime })\)  via Lemma  REF

From the NLC width \((k+2)\)  expression, construct a clique-width
\((2k+2)\)  expression for \(L(G^{\prime })\)  via Proposition REF  (The surplus edges
added during the chordal completion are removed at this step).

